Lectures in Logic and set Theory; vol. II: Set Theory
These two volumes present a basic course on mathematical logic and set theory. The contents are standard, including first order languages, axioms and rules of inference, basic metatheorems, the Löwenheim-Skolem theorems, Gödel's completeness theorem, the compactness theorem, and full proofs of both Gödel's incompleteness theorems in the first volume; the axioms of Zermelo-Fraenkel set theory, the axiom of choice, the natural numbers, partially ordered sets, ordinal and cardinal numbers with their elementary arithmetic, the constructible universe and elements of forcing in the second volume. Extreme care paid to details; most of the material is explained twice, first at an informal level and then formally; plenty of footnotes are added to sharpen the reader's understanding. This approach (together with a verbatim repetition of 57 pages from Volume 1 in Volume 2) may be appreciated by a student, who wishes to find every dot on every i. But it consumes a lot of space and consequently makes it impossible to include more advanced chapters: the rationals and the reals are not constructed, except for the Δ-system lemma, infinitary combinatorics is not introduced, the basics of Borel and of projective sets is missing and widely used principles, like MA, are never mentioned.